(1) As a density-based metric, we compute the posterior predictive density (PPD) [1]} following \(\exp {\frac{1}{N} \sum _{n= 0}^{N} \log ( \sum _{s}^{S} p(y_{n} \mid \theta ^{(s)}))}\) .\(y_{n} = y_{a}^{(t)}\)  or in the case of the 4-mode count tensor introduced in sec:data, \(n= (i,j,a,t)\)  is a multi-index such that \(y_{n} = y^{(t)}_{i\xrightarrow{}j}\) . PPD ranges between 0 and \(\infty \)  and measures the averaged log-likelihood of the held-out data given the inferred posterior parameters \(\theta ^{(s)}\) .
